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[Refractor] contradiction over ⊥-elim in trans∧tri⇒respʳ & trans∧tri⇒respˡ def #2657

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Mar 14, 2025
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[Refractor] contradiction over ⊥-elim in trans∧tri⇒respʳ & trans∧tri⇒respˡ def #2657

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[Refractor] contradiction over ⊥-elim in trans∧tri⇒respʳ & trans∧tri…
jmougeot Mar 12, 2025
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[Refractor] contradiction over ⊥-elim in trans∧tri⇒respʳ & trans∧tri…
jmougeot Mar 12, 2025
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contradiction over bot/elim
jmougeot Mar 13, 2025
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13 changes: 6 additions & 7 deletions src/Relation/Binary/Consequences.agda
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Original file line number Diff line number Diff line change
Expand Up @@ -8,14 +8,13 @@

module Relation.Binary.Consequences where

open import Data.Empty using (⊥-elim)
open import Data.Product.Base using (_,_)
open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_]′)
open import Function.Base using (_∘_; _∘₂_; _$_; flip)
open import Level using (Level)
open import Relation.Binary.Core
open import Relation.Binary.Definitions
open import Relation.Nullary.Negation.Core using (¬_)
open import Relation.Nullary.Negation.Core using (¬_; contradiction)
open import Relation.Nullary.Decidable.Core
using (yes; no; recompute; map′; dec⇒maybe)
open import Relation.Unary using (∁; Pred)
Expand Down Expand Up @@ -121,7 +120,7 @@ module _ {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} where
irrefl (antisym x<y y<x) x<y

asym⇒antisym : Asymmetric _<_ → Antisymmetric _≈_ _<_
asym⇒antisym asym x<y y<x = ⊥-elim (asym x<y y<x)
asym⇒antisym asym x<y y<x = contradiction y<x (asym x<y)

asym⇒irr : _<_ Respects₂ _≈_ → Symmetric _≈_ →
Asymmetric _<_ → Irreflexive _≈_ _<_
Expand Down Expand Up @@ -157,16 +156,16 @@ module _ {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} where
_<_ Respectsʳ _≈_
trans∧tri⇒respʳ sym ≈-tr <-tr tri {x} {y} {z} y≈z x<y with tri x z
... | tri< x<z _ _ = x<z
... | tri≈ _ x≈z _ = ⊥-elim (tri⇒irr tri (≈-tr x≈z (sym y≈z)) x<y)
... | tri> _ _ z<x = ⊥-elim (tri⇒irr tri (sym y≈z) (<-tr z<x x<y))
... | tri≈ _ x≈z _ = contradiction x<y (tri⇒irr tri (≈-tr x≈z (sym y≈z)))
... | tri> _ _ z<x = contradiction (<-tr z<x x<y) (tri⇒irr tri (sym y≈z))

trans∧tri⇒respˡ : Transitive _≈_ →
Transitive _<_ → Trichotomous _≈_ _<_ →
_<_ Respectsˡ _≈_
trans∧tri⇒respˡ ≈-tr <-tr tri {z} {_} {y} x≈y x<z with tri y z
... | tri< y<z _ _ = y<z
... | tri≈ _ y≈z _ = ⊥-elim (tri⇒irr tri (≈-tr x≈y y≈z) x<z)
... | tri> _ _ z<y = ⊥-elim (tri⇒irr tri x≈y (<-tr x<z z<y))
... | tri≈ _ y≈z _ = contradiction x<z (tri⇒irr tri (≈-tr x≈y y≈z))
... | tri> _ _ z<y = contradiction (<-tr x<z z<y) (tri⇒irr tri x≈y)

trans∧tri⇒resp : Symmetric _≈_ → Transitive _≈_ →
Transitive _<_ → Trichotomous _≈_ _<_ →
Expand Down